On Causal Structure in (1 + 1)-Dimensional and (2 + 1)-Dimensional Generalised Causal Dynamical Triangulations Scott Nicholas Allan Smith Department of Theoretical High Energy Physics Radboud University Nijmegen Supervisor: Prof. Renate Loll February 28, 2014 Acknowledgements First and foremost I owe a great many heartfelt thanks to my supervisor Prof. Renate Loll. Without her many hours of tireless advising and checking, my own understand- ing of the subject would still be unbearably murky and certainly this piece of work could not have come to be at all! I would like to thank Samo Jordan, Ben Ruijl, Rene´eHoekzema and Luis Pires for advice and discussions at various stages of the project which resolved my misunder- standings and helped me to sharpen and advance my thoughts and ideas. I also thank Nuffic for their financial support during my whole study period in the Netherlands through the vector of their Huygens Scholarship program. Finally, last but most assuredly not least, I thank my parents Stephen & Jacqueline Smith and my sister Claire Smith for their unwavering support in this intellectual adventure and indeed all of my endeavours. 1 Contents 1 Introduction and Motivation 9 1.1 Introduction................................ 9 1.1.1 Causal Dynamical Triangulations . 9 1.1.2 Generalised Causal Dynamical Triangulations . 15 1.1.3 Causalconsiderations. 18 1.2 Curvaturein(g)CDTmodels . 19 2 CurvatureandPhotonPropagationin(1+1)d 22 2.1 Introduction................................ 22 2.2 LorentzAngles .............................. 26 2.3 WilsonLoops ............................... 32 2 2.3.1 Introduction............................ 32 2.3.2 CoordinateFramesaroundaVertex. 36 2.3.3 Lorentz Transformations in gCDT Triangulations . 37 2.3.4 Single Vertex Wilson Loops . 42 2.4 PhotonPropagation ........................... 43 2.4.1 Introduction............................ 43 2.4.2 PhotonTrajectories. 44 2.4.3 EntryandExitPoints ...................... 47 2.4.4 Example Application of Algorithm . 53 2.4.5 Path of γ1 ............................. 57 2.4.6 Path of γ2 ............................. 61 2.4.7 FinalSeparation ......................... 64 3 Vertex Causal Structure in (2+1)-Dimensional gCDT 65 3.1 Introduction................................ 65 3.2 StructureAboutTheVertex . 69 3 3.2.1 Fundamental Spacelike Structures . 69 3.2.2 Relationships between Planes and Spirals . 78 3.2.3 The Causal Structure Associated with a Stack of Planes . 85 3.2.4 The Arrangement of Stacks of Planes about a Vertex . 100 4 Contractible Closed Timelike Curves in (1+1)-Dimensional CDT 106 4.1 Introduction................................ 106 4.1.1 A Useful Restatement in (1+1)-dimensions . 107 4.2 Spacelike Objects in the gCDT Triangulation . 108 4.2.1 Introduction............................ 108 4.2.2 Preliminary Definitions . 108 4.2.3 CDT-like Initial Boundary . 118 4.2.4 Non CDT-like Initial Boundary . 126 4.2.5 Sub-areasoftheTriangulations . 143 4.3 TransformationsBetweenTriangles . 152 4.4 The Structures and Transformations Associated With A Timewalk . 159 4 4.4.1 PathsThroughtheTriangulations. 159 4.4.2 An Algorithm to Shrink The Path . 166 5 Discussion 174 5.1 LocalCurvaturein(1+1)Dimensions . 174 5.2 Global Causal Structure in (1 + 1)-Dimensional and (2 + 1)-Dimensional gCDT ................................... 176 A Representative Vertices 179 B Useful Triangle Properties 183 B.1 TriangleMetrics.............................. 183 B.1.1 Down ............................... 183 B.1.2 Left ................................ 183 B.1.3 Right................................ 184 B.2 TriangleVolumes ............................. 184 B.2.1 tts ................................. 184 B.2.2 sst ................................. 185 5 B.3 EquationsforTriangleEdges. 186 B.3.1 Up................................. 186 B.3.2 Down ............................... 186 B.3.3 Left ................................ 187 B.3.4 Right................................ 187 C Lorentzian Angle Results 188 C.1 InnerProducts .............................. 188 C.2 Cosines and Sines For General α ..................... 189 C.3 Cosines and Sines For α =1....................... 190 C.4 Angles................................... 191 C.5 DeficitAngles ............................... 192 D Determination of Sign for Sorkin Angles 194 E Wilson Loop Calculations 196 E.1 Relation between Coordinates in Left & Right sst Triangles and Stan- dardMinkowskiCoordinates . 196 6 E.2 Lorentz Transformation in the Left and Right sst Triangles . 197 E.3 LorentzTransformationProperties . 198 E.4 ArrangementsofAdjacentTrianglesingCDT . 201 E.5 Lorentz Transformation Angles for Adjacent Pairs . 204 E.6 Single Vertex Wilson Loop Results . 204 F Photon Propagation Algorithm and Results 206 F.1 EntryandExitPoints .......................... 206 F.1.1 DownttsTriangle......................... 207 F.1.2 LeftsstTriangle ......................... 208 F.1.3 Right sst Triangle . 209 F.2 CoordinateTransformations . 210 F.3 PhotonDeflections ............................ 211 G Explanation of Hoekzema’s Proof 213 H Notes from A Meeting to Discuss gCDT Causal Structure 223 7 I Example Triangulations 226 8 Chapter 1 Introduction and Motivation 1.1 Introduction 1.1.1 Causal Dynamical Triangulations Already in 1916, merely one year after the introduction of general relativity to the physics community by Einstein [1], Einstein himself observed that the theory of gen- eral relativity would require quantum corrections [2]. In the meantime, all three other fundamental forces have been quantised with a great deal of success and correspond- ingly a complete theory of quantum gravity has continually been sought [3]. Whilst there have been a great number of ideas, from those perturbative approaches which quickly led to uncontrollable divergences [4], to the latest ideas which include string theoretical approaches [5], loop quantum gravity [6] and various other approaches (for example causal sets [7], non commutative geometry [8]), there is still a great deal of work to do in order to achieve this goal. Generally the approaches fall into two categories - those which start from a canonical (or Hamiltonian) formulation of gravity, and those which start from a covariant (or 9 path integral) formulation of gravity. The latter begin at the path integral Z = D [g] eiSEH [g] (1.1) Z with d [g] representing an integral over all metrics, and SEH the Einstein-Hilbert action, and attempt to obtain useful information from it, including for example R observables and propagators. This has unfortunately proved a tough nut to crack, and several of the above mentioned schemes are geared towards exactly this goal. Causal Dynamical Triangulations (CDT)1 is one such scheme. There is, as yet, no set of tools amenable to performing calculations with this object which is valid across the entire range of physical parameters (for example, energy) in which there might be interest. CDT proposes a calculational framework which takes the form of a certain regularisation of (1.1). This is implemented by “integrating over” a restricted set of metrics (i.e. geometries). The chosen set of geometries is the set of triangulations, which are a class of piecewise linear simplicial complexes. Simplicial complexes can be thought of as geometries obtained by gluing together d-dimensional polytopes according to certain sets of gluing rules [10]. In this thesis, as is standard in the CDT literature, a d-dimensional simplicial complex where all 2-dimensional simplices have exactly three sides (that is, all faces are triangles) will be called a d-dimensional triangulation [11]. 1 iSRegge[T ] This choice leads to a nonperturbative path sum Z = T C(T ) e , which lends itself more easily to calculation. Here T stands for all triangulations of given dimen- P sion d, C(T ) is the order of the symmetry group of the triangulation, and SRegge is the action for piecewise linear spacetimes given by Regge [12]. In the CDT approach, yet stronger restrictions are imposed upon the triangulations used. The length of each edge of a simplex is allowed to have one of only two possible values for its squared length - a2 for spacelike edges (called “s” edges in this thesis), and αa2 for timelike edges (called “t” edges in this thesis), where − 1This subsection is based upon [9], except from the parts to which another citation is explicitly attached. 10 a2,α> 0 and metric signature ( , +, +, +) is assumed). This places in turn strong restrictions upon the kinds of d−-dimensional simplices that are allowed to occur in the d-dimensional triangulation. Without this restriction, extra divergences are introduced to the path integral and the risk of overcounting is run. sss ttt tts sst Figure 1.1: The four triangles that that may be constructed from the allowed side lengths in CDT. The convention used here, blue lines for spacelike (s) edges and red lines for timelike (t) edges, shall be used throughout this thesis. Additionally, green shall stand for a timelike line. CDT further restricts the set of allowable simplices to only those which are Lorentzian. This gives rise to the “causal” part of the name CDT, since by choosing to use ex- clusively simplices with such a property the consideration and definition of causal structure and causality on the triangulations is facilitated. In the simplicial com- plexes employed here, the space inside the d-simplices is chosen to be flat, hence the simplices allowed in CDT contain Minkowski space. This does not however prevent a CDT triangulation from including curvature,